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Statistical analysis of significance estimation... 1

UNIWERSYTET TECHNOLOGICZNO-PRZYRODNICZY IM. JANA I JĘDRZEJA ŚNIADECKICH

W BYDGOSZCZY

ZESZYTY NAUKOWE SCIENTIFIC JOURNAL 268

TELEKOMUNIKACJA I ELEKTRONIKA

21

MODELING DYNAMICAL SYSTEMS

BYDGOSZCZ – 2018

2 Vasyl Zaiats, Olga Rybytska, Mariy Zaiats

REDAKTOR NACZELNY prof. dr hab. inz. Józef Flizikowski

REDAKTOR NACZELNY SERII dr inz. Beata Marciniak

REDAKTOR NAUKOWY prof. dr hab. inz. Vasyl Zaiats

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Wydawnictwa Uczelniane Uniwersytetu Technologiczno-Przyrodniczego Bydgoszcz 2018

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współfinansowanego ze środków Europejskiego Funduszu Rozwoju Regionalnego w ramach Regionalnego Programu Operacyjnego

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ISSN 1899-0088

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International Editorial Board:

Editor in Chief: Zaiats Vasyl, Department of Computer Science ACCS Institute NUWEE, Rivne, Ukraine, Faculty of Telecommunications, Computer Science and Electrical Engineering, UTP University of Science and Technology, Bydgoszcz, Poland.

Deputies Chief Editor:

Kiedrowski Piotr, Faculty of Telecommunications, Computer Science and Electrical Engineering, UTP University of Science and Technology, Bydgoszcz, Poland. Pentiuc Stefan – Gheorghe, Research Center in Computer Science from the of Suceava University “Stefan cel Mare", Suceava, Romania. Yavorsky Igor, FMI by G.V. Karpenko of NASU, Lviv, Ukraine, Faculty of Tele-communications, Computer Science and Electrical Engineering, UTP University of Science and Technology, Bydgoszcz, Poland.

Members of the editorial board:

Bomba Andriy, Department of AM and Informatics of the National University, Rivne, Ukraine. Hahanov Vladimir, CIM KNURE, Kharkiv, Ukraine. Hartnett Michael, National University of Ireland, Galway. Kirilenko Alexander, Academician of NAS of Ukraine, Institute of Cybernetics, Kyiv, Ukraine. Martyniuk Petro, Department of AM, NUVEI, Rivne, Ukraine. Moshynskyi Viktor, NUVGP, city of Rivne, Ukraine. Pasichnyk Volodymyr, Department of ICM NLPP, Lviv, Ukraine. Popa Valentin, Stefan cel Mare State University, Suceava, Romania. Savula Yarema, Department of Applied Mathematics of Lviv National University named after I. Franko, Lviv, Ukraine. Samoilenko Anatoliy, National Academy of Sciences of Ukraine, IMA NASU, Kiev, Ukraine. Semenets Valery, KNURE, Ukraine. Slyusarchuk Vasyl, NUWEE, Rivne, Ukraine. Zgurovsky Michael, Academician of the National Academy of Sciences of Ukraine, National Technical University of Ukraine "KPI", Kyiv, Ukraine.

Responsible editor of the magazine:

Rybitskaya Olga, Department of NU "LP", Lviv, Ukraine. Magazine's responsible secretaries:

Zakrzewski Zbigniew, Faculty of Telecommunications, Computer Science and Elec-trical Engineering, UTP University of Science and Technology, Bydgoszcz, Poland. Shatnyi Sergeii, Department of NUWEЕ Rivne, Ukraine. Turcu Cristina Elena, Suceava University “Stefan cel Mare”, Suceava, Romania. Marciniak Beata, Faculty of Telecommunications, Computer Science and Electrical Engineering, UTP University of Science and Technology, Bydgoszcz, Poland.

Publishing editorial board: Burov Yiwgen, Department of ISN NU "LP", Lviv, Ukraine. Chumachenko Svitlana, Design Automation Department KNURE, Kharkiv, Ukraine. Vyklyuk Yaroslav, Science and International Relations, Private Bukovinsky University, Chernivtsi, Ukraine. Gaitan Vasile Geeorghita, Stefan cel Mare State University, Suceava, Romania. Krivulya Gennady, Department of ACT KNURE, Kharkiv, Ukraine. Marciniak Tomasz, Faculty of Telecommunications, Computer Science and Electrical Engineering, UTP University of Science and Technology, Bydgoszcz, Poland. Majewski Jacek, Faculty of Telecommunications, Computer Science and Electrical Engineering, UTP University of Science and Technology, Bydgoszcz, Poland. Saphonik Andriy, NUVEE, Rivne, Ukraine. Tevyashev Andriy, AM, KNURE, Kharkiv, Ukraine. Turbal Yuriy, Department of AM NUVEE, Rivne, Ukraine.


Contents

Introduction ...................................................................................................................... 7

Vasyl Zaiats, Olga Rybytska, Mariy Zaiats – Statistical analysis of significance estimation and quantity of the information in mass service ..............................................................................................................................9

Andrii Bomba, Mykhailo Boichura – Spatial analogues of numerical quasiconformal mapping methods for solving identification problems ......................... 25

Gennady Krivoulya, Vladislav Shcherbak – Intellectual wireless diagnosis of linearly distributed objects in real time ...................................................... 35

Oksana D. Kozhushko, Petro M. Martyniuk – Computer modelling of the dynamic system «porous medium – moisture – chemical substance» in the case of soil desalinization by rainfalls ............................................... 47

Myroslav I. Stodilka, Oleksandra A. Baran – Peculiarities of convective flows formation in the solar photosphere ....................................................................... 61

Andriy Tevyashev, Switlana Iievlieva – Information and analytical technology for control and operation management of gas transportation systems operation modes ................................................................................................ 69

Sergiy Babych, Yuriy Turbal – Methods of the schedule matrix forming based on the modified permanent ................................................................................... 85

Lista recenzentów – Reviewers list ................................................................................ 93


INTRODUCTION

In the hands of the Reader, we give a special issue of Scientific Journal devoted to Modeling dynamic systems. This topic is very timely, taking into account the use of real-time systems in many areas of life.

The issue contains seven articles describing ways of modeling systems in various fields of knowledge, presented below.

Thematic scope of articles: 1. Modeling and software of application applied. 2. Theory of information, coding and signal processing. 3. Numerical methods in applied applications. 4. Deterministic and stochastic dynamical systems and its optimization. 5. Intelligent data processing systems and solutions (by branches of knowledge). 6. Innovative technologies and machine learning in science, education and pedagogy. 7. Applied problems of natural sciences (mathematics, physics, chemistry, biology,

geology, ecology, electrical engineering and electronics, genetics and medicine, transport and construction) in information technologies.

We hope that the presented approach to the implementation of tasks will interest

the Reader and contribute to the exploration of the presented thematic scopes of articles.

Vasyl Zaiats Beata Marciniak

UNIWERSYTET TECHNOLOGICZNO-PRZYRODNICZY IM. JANA I JĘDRZEJA ŚNIADECKICH W BYDGOSZCZY

ZESZYTY NAUKOWE NR 268 TELEKOMUNIKACJA I ELEKTRONIKA 21 (2018) 9–23

STATISTICAL ANALYSIS OF SIGNIFICANCE

ESTIMATION AND QUANTITY OF THE INFORMATION IN MASS SERVICE

Vasyl Zaiats1,2, Olga Rybytska3, Mariy Zaiats3

1UTP University of Science and Technology, Bydgoszcz, Poland

e-mail: [email protected] 2National University of Water Management and Natural Resources, Rivne, Ukraine

e-mail: [email protected] 3National Lviv Polytechnic University, Lviv, Ukraine

e-mail: [email protected], [email protected]

Summary. On the basis of the theory of images, a new approach that allows quanti-fying the significance and amount of information is proposed. The approaches to the definition and calculation of the basic concepts of information theory, in particular, the amount of information and evaluation of its significance, are based on statistical considerations (classical approach), algorithmic theory (algorithmic approach), and the theory of pattern recognition (figurative approach). Approaches to the pro-cessing of fuzzy information are proposed in the conditions of incomplete determi-nation of the vector of input characteristics, are based on the theory of fuzzy sets and fuzzy measures. Their analysis have been carried out, the limits of their use, and areas of effective application, in particular, regarding mass service systems.

Keywords: amount of information, significance of information, fuzzy sets and fuzzy

logic, systems of mass service

1. INTRODUCTION AND THE PURPOSE OF THE STUDIES

The significance of theoretical results, as a rule, is based on their practical value in applied applications that directly or indirectly confirm the feasibility of their application and do not violate the logic of reasoning. Newly introduced terms or analytical or de-scriptive formulas that are not operational and do not have practical value over time are excluded from scientific consideration.

The purpose of the work is to show the universality of the approach to quantitative evaluation of the value of information on the basis of the theory of images, to compare it with existing methods and approaches to assessing the value of information, to indi-cate the limits of practical applications, to determine the diradvantages and disad-vantages, the directions of further research.


2. MAIN RESULTS

Getting information about the system is associated with a decrease in its entropy. How much the entropy of the system will decrease, so much the information about it will grow. From this perspective, information can be defined as a negative entropy of the system.

This point of view is formulated as the non-aggressive principle of information in the [1]. There was in the papers [4, 5, 8, 14] proposed approaches to the estimation of the

quantity and value of information on the basis of the classical theory of information, of the theory of pattern recognition, fuzzy sets and measures. They considerably extend the area of applicationsin comparison with classical results [3, 10].

2.1. APPROACH TO EVALUATION OF QUALITY AND VALUE OF INFORMATION

The classical theory of information, developed by C. Shannon [10], today is a per-fect universal apparatus for solving problems related to information encoding, its trans-formation and optimal transmission over long distance communication channels. The limitation of this theory is that it in no way takes into account the semantics (mode of formation) of information and totally ignores the human factor in the formation of in-formation, that is, the notion of value of information is largely ignored, which is largely a subjective characteristic and depends on the purposes and preferences of the user. Obviously, this is the fee for the ability to calculate the amount of information based on statistical considerations (probability characteristics), regardless of its quality and value.

Note that the amount of useful information is always magnitude positive. In the case of the mutual influence of one system on the other, there appears conditional un-certainty of the states of the system, which may lead to a change in the sign in determin-ing the amount of information. Then we deal with another quality of information – misinformation or false information.

Algorithmic theory, based on the notion of complexity of the algorithm, has made a significant step in the direction of taking into account the method of formation of information. Despite a number of shortcomings of this theory, its main provisions are used in the figurative concept of information theory [4, 8]. According to this concept, the information should be understood as recognizable images that are stored in the memory of a computer or any other cybernetic machine. The image is a signal recorded in the sensor memory of the scanner devices of the cybernetic machine. Thus, in order to obtain information, the procedure for recognizing the input object, the image based on its mapping Q – of a certain standard of this image, created on the basis of arrange-ments between the sender and the message receiver, should be implemented.

The amount of information I0 contained in a certain image obtained and success-fully recognized by a cybernetic machine can be determined by the formula:

10 [ ( )]I q F G O

-= (1)

where q – the probability of correct recognition of the image; F – the function of obtaining the length of the extended (taking into account the cycles) of the image recognition program; G – the length of the program of recognition of the image, expressed in bits, O – display (image) of the real object.

Obviously, in approach (1), it is possible to calculate the amount of information of complexly structured images, which can be separate words, sentences or texts or fig-ures. In this case, the amount of information will depend significantly on the length of


the extended recognition program, the probability of correct recognition and the func-tionality of the cybernetic machine that implements the recognition procedure. For the objective calculation of the amount of information according to formula (1), the follow-ing should be taken into account: a) image recognition programs should be optimal for their own size, performance and

functionality; b) an increase in the number of operations (commands) that must be performed by the

machine for successful recognition, leads to an increase in the information we obtain from the image;

c) the smaller the probability of the correct recognition of the image, the greater the amount of information it has;

d) the amount of information in an insignificant message is equal to the length of this mes-sage, expressed in bits, multiplied by the length of the program of recognition of one bit of this message and divided by the probability of correct recognition of the message.

The first experiments [8] confirmed the right to exist for such an approach to cal-

culating the amount of information, although its effectiveness can only be realized in the process of producing numerical (computer) experiments with the recognition of complex structured objects. Such an approach has the following disadvantages: a) to a large extent, a subjective estimate of the amount of information that carries an

image due to both the technical capabilities of the cybernetic machine and the quality of the recognition program;

b) the property of the additivity of the amount of information that is present in the classical information theory is lost;

c) formula (1) is too cumbersome, which complicates the carrying out of analytical assessments.

The second disadvantage when calculating the amount of semantic (figurative)

information is eliminated if you go to the logarithmic scale:

0 2 2- ( )]I Log q Log f O= + (2)

where f is the length of the pattern recognition program, expressed in the number of operations (machine commands) needed to successfully recognize the image.

2.2. THE PRINCIPLE OF COMPUTING VALUES OF INFORMATION ON THE TARGET FUNCTIONS BASIS

With regard to the value of information, then it can be said about it if it is necessary to achieve a certain goal after receiving information by the user, that is, ensuring the implementation of the target function. In work [4] AA Harkevich proposed the value of information to be calculated as:

02

i

pF Log

p= (3)

where p0 – the probability of correct solution of the problem to obtain information, and – the probability of correct solution of the problem after receiving information. This approach has the right to exist, although it raises a number of doubts as to its effectiveness and feasibility. First, the unit of measurement of the value of information in this approach is a bit, as in the case of deducting the amount of information. Obviously, when a new value is introduced, a new dimension or calculation should be made in dimensionless


units. Secondly, formula (3) can not claim objectivity, because the estimates p0, p1 will be made by the user. Third, the methodology for evaluating these probabilities is not obvious. Fourthly, the value of information is a dynamic value [7] and as information arrives, it should change. Obviously, when calculating the value of information in the role of the target function to achieve the goal, you must choose the satisfaction of certain needs of the user (material, spiritual, aesthetic, taste, cognitive, etc.) or to perform certain actions.

The most expedient considering the diversity of the needs of the user is an approach in which the value of information will be calculated as a percentage: 100% – provided the value of information; 0% – provided that the target function is not reached. Thus, the value of the information F for the current value of the function of the goal Z and the achieved efficiency E after receiving the message at the given time can be specified as a rule:

100%, 00%, 0

if Z EFif Z E

- ==

- > (4)

In order to increase the accuracy of the calculation of the value of information it is advisable to have the formed target function and at the next moments of time and to expand the scale of the calculation of the value of information with a certain step. The first condition significantly increases the value of the information after receiving the entire message, and the second can be realized by the formula:

x 100%EFZ

= (5)

Formula (5) is a refinement (4), which is confirmed by the consideration of specific examples of consideration of mass service systems [9]. The obtained results testify to the expediency of the applied application of the proposed approach to the quantitative assessment of the value of information, regardless of its content, nor the way of submis-sion. In the example of the mass service system, it is shown that if formula (4) allows estimating the turnover of a commodity and its value per year, then (5) provides scaling of the process with an arbitrary step, which allows to optimize the process of release of goods and to control the financial costs after almost every financial operations.

2.3. ESTIMATES OF QUALITY AND VALUE OF INFORMATION IN MASS MEDIA SYSTEMS

2.3.1. Regulatory approach

A fundamentally different approach to determining the quantity and value of in-formation proposed by the Russian scientist AM Kolmogorov [5]. His algorithmic theo-ry of information is based on the notion of the complexity of the algorithm of trans-forming one object to another. In this approach, it is crucial to establish a mutual rela-tionship between the objects being studied and the length of the program they are pro-cessing. The amount of information on the theory of algorithms for the conversion of one object to another is defined as the length of the program, which enables the conver-sion of object A to object B:

[ ( , )]I f G A B= (6)

where G is a program for converting object A into object B; f is a function that defines the length of the conversion program in bits.


Note that the amount of information for such an approach is significantly depend-ent on the choice of the structural element of the transformation. The number of infor-mation is maximal if the pixel is selected as a conversion element, and the minimum is when the element of the whole letter is selected as the element of conversion, and it will be intermediate when selecting the part of the letter by the conversion element. The same amount of information for a particular letter is impossible to calculate, since two objects need to be implemented for the algorithm (8). Comparing the same letter with it itself has no content, because the length of the conversion program in this case will be zero.

Thus, the following drawbacks are evident in the algorithmic approach: a) when calculating the amount of information as a parameter, the length of the conver-

sion program is used, which essentially depends on the structure of the elements that make it possible to convert one object to another. The smaller the selected structure of the elements, the longer becomes the conversion program for the same objects;

b) the same conversion program can be used to process a whole set of objects, the analysis of which requires the execution of a different number of computer com-mands, while the length of the program remains unchanged;

c) there is no method for measuring the amount of information when considering a single object, since according to formula (4) they require two;

d) the property of the additivity of the information is lost when considering interacting systems, which considerably complicates their analysis.

2.3.2. Semantic approach These shortcomings are also characteristic of the semantic approach in the calcula-

tion of information based on the elaboration of logical statements:

2 ( )I Log L O= (7)

where L is a function that depends on the number of states in logical statements (varia-bles, facts, rules); O – logical statement or predicate.

The calculation of the amount of information in individual letters or words for the approach (7) is impossible in principle, since neither individual letters nor individual words are logical statements.

2.3.3. The approach based on the fuzzy sets and measures theory The aforementioned disadvantages are eliminated in the approach based on the ap-

plication of methods of recognition theory and operates with the notion of information, the quantity and value of information that are different from the classical definitions obtained on the basis of probability theory.

The problems of mathematical and computer modeling today lie, in particular, in the impossibility of applying precise logic rules and models of tasks with clearly defined input parameters in cases where, for some reason, there are contradictions, uncertainties or uncertainties of information about the object being studied, system or phenomenon [15].

Uncertainty is known to arise due to a lack of knowledge relating to a specific event [7]. She is present to the experiment. The mathematical model of uncertainty is based on the theory of probabilities, the theory of possibilities, confidence measures, the theory of prediction and prediction, and a number of others.


The phenomenon of fuzziness arises in the process of combining into one whole objects that have a common property :

ownsX x x

where x all the elements of a certain universal set run.

Given that there are always elements x in the reality that it is unclear whether they possess the specified property or not, X it is not a plural in the classical sense. Any at-tempt to interpret the general description leads to fuzzy concepts, since the exact de-scription contains an excess of details. Increasing the accuracy of the description leads to an increase in the amount of information, the content content of which decreases until the time when the accuracy and meaningfulness do not become mutually exclusive. For the first time, LA stressed the need for uncertainty for the transmission of content. Zade [13]. It was the ideas of this American scientist who pushed for the development of "fuzzy mathematics" [6], which, along with the apparatus of fuzzy sets, contains other methods of work with uncertainty.

The application of the theory of fuzzy sets and measures is a step towards the con-vergence of the precision of classical mathematics with a false inaccuracy of the envi-ronment, an attempt to overcome the linguistic barrier between a person whose judg-ment and evaluation are approximate and fuzzy, and technical means which can only carry out precise instructions [11].

A device that allows you to work with fuzzy logic, "blurred" parameters of models, is a Fuzzy-technology device. The Fuzzy-Technology division has fuzzy expert systems.

Linguistic (descriptive) variables extend the ability to represent knowledge. They are determined by fuzzy sets whose values are established by membership functions. Mem-bership functions can be obtained through subjective expert assessments [12], or by ana-lyzing fuzzy clusters. According to [12], fuzzy expert systems can be implemented when the cost of acquiring accurate information, that is, information in absolute terms, exceeds the maximum revenue from the restructuring of a model or is virtually impossible.

It is known that the initial stage of constructing artificial intelligence on the basis of the use of natural language is based on ambiguous logic and the mechanism of output with rigid rules.

The modern or second generation of expert systems possesses at least two peculiari-ties: fuzzy presentation of knowledge and fuzzy deductions. One of the most common problems of logical derivation in the conditions of fuzziness can be formulated as follows:

( ) log " , ". ( ). ?

Given fuzzy ical rule If A then BObserved А A to some extent What should be B

After receiving the fuzzy set of conclusions, find a specific numerical match

(conduct dephasing). Consider the object with one output and n inputs of the form:

1 2( , ,..., ),ny f x x x= (8)

where the set of values 1,..., nx x of input variables; y – output variable. To construct a mathematical model on the basis of establishing the relationship

between input and output variables in accordance with experimental data, by conducting a phasing operation, quantitative and qualitative variables are translated into linguistic terms:


, , 1, , i i iU u u i n (9)

,Y y y (10)

where ,i iu u – the smallest and the highest possible value of the variable ;ix ,y y – the smallest and the highest possible value of the output variable y.

To solve the problem (8), it is necessary to apply a method of making a decision

by which the fixed vector of input variables 1 2 * * * * *n i ix x ,x ,...,x , x U would unambiguously be placed in accordance with the solution Yy * . For the formal establishment of this dependence we shall consider the input nixi ,1 , and output y parameters as linguistic variables given on universal sets (9), (10). To evaluate linguistic variables nixi ,1 , and y, we will use qualitative terms from the following term sets:

1 2, ,..., ipi i i iA a a a term-set of input variable nixi ,1 , ; mdddD ,...,, 21 term set of output variable y. To construct term sets it is possible to apply, for example, the method proposed in [15].

For each term of each linguistic variable, based on expert knowledge, the

memberships functions ( )pia x and ( )jd y (trapezoidal, triangular, rectangular,

sinusoidal, parabolic, etc.) [13] are constructed based on expert knowledge, where

( )pia x – the degree of belonging of the element ix U to the term

, 1, ; 1, ; jdpi i ia A i n p p y the degree of belonging of the element y Y to the term .,1 , mjDdj

The definition of linguistic estimates of variables and the membership functions necessary for their formalization is the first stage in the construction of a fuzzy model of the object being studied. In the literature on fuzzy logic, he received the name of the fuzzification of variables [16].

The next step is to create a fuzzy knowledge base. Let the object (8) know the rules that connect its inputs and output using vectors

such as:

1 2, ,..., , , 1, k nV x x x y k N , and ,.......1 mj kkkN (11)

where jk the number of experimental data corresponding to the same value jd of the term-set of the output variable y; m – the total number of terms of the output variable, and in the general case ....1 mkk

We will assume that the number 1 2 ... , nN p p p of available experimental data is less than the total overview of various combinations of possible terms of input variables

, 1, .ip i n . Then the knowledge base is a table formed according to the following rules:


1. The table's size is equal ( 2), N n where n + 2 – number of columns and N – the number of rows;

2. Each line of the matrix is a combination of input variables assigned by the expert to one of the possible values of the term-set of the output variable y. In this case, the first k1 lines correspond to the value of the output variable 1y d= , the following k2 lines – to the value 2dy , etc., and the last km lines – to the value ;mdy

3. The first n columns of the matrix correspond to the input variables ;,1 , nixi the )1( n -th – the weight ,,1 ,,1 , jjp kpmjW of the rule and )2( n -th – the value

to the output term-set dj of the variable ,,1 , mjy corresponding to the combina-tion of values in the first )1( n -th columns.

4. The element ,jpia located at the intersection of the i-th column and the jp – line corresponds to the linguistic evaluation of the parameter xi in the row of knowledge matrix with the number Pj . In this case, the linguistic assessment

jpia is chosen

with the term-set corresponding to the variable ,ix i.e. .,1 ;,1 ;,1 , jijp

i kpmjniAa

When an expert creates linguistic rules such as "IF - THEN" that form the basis of fuzzy knowledge about a particular object, the expert's confidence in each rule may be different. If one rule in the opinion of an expert can serve as an undeniable truth, then according to another rule in the same expert there may be some doubts.

In order to reflect these different degrees of confidence in the base of fuzzy knowledge, the weighting of the rules is introduced - these are numbers from the inter-val [0, 1] that characterize the expert's confidence in each particular rule chosen by him to make a decision. The general view of the knowledge base is given in Table 1.

Table 1. General view of the fuzzy knowledge base

Incoming combination number Input variables Weight Output variable

1x ...... ix nx w y 11 111a

11ia

11na 11w

1d 12 121a

12ia

12na 12w

… … … … … … 11k

111

ka 11k

ia 11k

na 11 kw … … … … … … …

1j 1

1ja

1jia

1jna 1jw

jd 2j

21ja

2jia

2jna 2jw

… … … … … … jjk

jjka 1 jjk

ia jjk

na jjkw

… … … … … … …

1m 1

1ma

1mia

1mna 1mw

md 2m

21ma

2mia

2mna 2mw

… … … … … … mmk

mmka1 mmk

ia mmk

na mmkw


After building the knowledge base, you need to carefully check in Table 1 the presence of the opposite in the content of the lines, that is, rules that the same input variables have different output values. The introduced matrix of knowledge defines a system of logical utterances such as "IF - THEN, ELSE", which associate the values

nxx ,...,1 of input variables with one of the possible output values :,1 , mjdj

IF 1111 ax AND 1122 ax AND … AND 11nn ax (with weight 11w ), OR 1211 ax AND 1222 ax AND … AND 12nn ax (with weight 12w ),

OR … OR 1111 kax AND 1122 kax AND … AND 11knn ax (with weight 11 kw ),

THEN ,1dy OTHERWISE IF 2111 ax AND 2122 ax AND … AND 21nn ax (with weight 21w ), OR …

OR 2211 kax AND 2222 kax AND … AND 22 knn ax (with weight 22 kw ), THEN ,2dy OTHERWISE …

IF 111 max AND 122 max AND … AND 1mnn ax (with weight 1mw ), OR …

OR mmkax 11 AND mmkax 22 AND … AND mmknn ax (with weight mmkw ), THEN .mdy

A similar system of logical expressions is called a fuzzy knowledge base. Using the operations (OR) and (AND) described system of logical statements can be rewritten in a more compact form:

( )1 1

, 1, .jk n jp

i i jp i

x a y d j m= =

é ùê ú= = =ê úë û

(12)

Thus, the input relation (8), which establishes the connection between the input pa-rameters nixi ,1 , and the output variable, is formalized in the form of a system of fuzzy logical statements (12) based on the created matrix of knowledge. The rules of the described fuzzy system, in which the degree of truth is different from zero, is consid-ered to be active.

In [9], a method is proposed to use fuzzy logic equations that are based on a knowledge matrix or isomorphic system of logical expressions (12) and allow us to calculate the values of the membership functions of the output variable for the fixed values of the inputs of the object.

Linguistic estimates jpia of the variables 1,... nx x , contained in the logical state-

ments (12) will be considered as fuzzy sets defined on universal sets (9). We introduce the following notation:

)( ia x

jpi the membership function of the parameter ix to the fuzzy term

;,1 ,,1 ,,1 , jjp

i kpmjnia nd xxxj ,..., 21 membership function of the vector of the input variables nxxxx ,..., 21 of the term of the output variable .,1 , mjdy j


Thus, we have two types of functions, the relationship between which is deter-mined by the base of fuzzy knowledge (12), on the basis of which you can output a system of logical equations, which can be submitted in a compact form:

1 2 1 1( , , ..., ) ( ) , 1. .

j jpj i

k nd an jp ip i

x x x w x j mm m= =

æ öé ù÷ç ê ú÷= =ç ÷ç ê ú÷çè øë û (13)

where is the logical OR; logical "AND". The decision },,...,,{ 21* mdddDd that corresponds to a fixed vector of the values

of input variables * * * *1 2, ,..., nx x x x will be carried out in accordance with the following algorithm constructed using the apparatus of the fuzzy (blurry) logic [9]: 1. The possible range of change of controlled parameters is determined, a knowledge

base is created with the use of expert data and a system of fuzzy logic equations is derived (13).

2. The vector of the values of the input variables * * * *1 2, ,..., nx x x x is fixed. 3. Specifies the function of fuzzy term membership for different controlled parameters. 4. Using logical equations (13), the values of many parametric membership functions

),....,,( **2*1 n

d xxxj of vector X for all values mjdj ,1 , of the output variable y are calculated. In this case, the logical operations (OR) and (AND) over the mem-bership functions are replaced by the operations max and min:

( ) ( ) max[ ( ), ( )],a b a bm m m m = (14) ( ) ( ) min[ ( ), ( )].a b a bm m m m = (15)

That is, first find the minimum values of belonging functions in each rule, and then among them they choose the highest value of the membership function among all rules for each value mjdj ,1 , , which corresponds to the original variable y. Thus, the conclusion is made that the origin variable y belongs to a term ,*jd whose membership function is maximal.

The proposed algorithm uses the idea of identifying the linguistic term by the max-imum of membership function and generalizes this approach to the entire knowledge matrix. The computational part of this algorithm is easily realized by simply applying operations max and min.

To obtain a clear number from an interval ],[ yy that corresponds to a fuzzy value of the output variable, it is necessary to apply a dephasing operation. You can define this clear number, for example, by the gravity method:

( )

( )

Max

MinMax

Min

y y dyy

y dy

(16)

where Min and Max is the left and right points of the interval of the fuzzy set of the source variable y.


2.3.4. Results of statistical analysis on an example of a system of mass service

The task of forecasting the value of unrealized goods of a certain type at the end of the trade season of a trading firm dealing with chemical protection products of plants is considered in the paper. Obviously, minimizing the balance from unrealized goods will increase the efficiency (productivity) of the trading firm.

The experts found that the weighting factors that affect the current balance Y are: x1 ("balance") – the balance of the previous trading season (in US dollars); x2 – "New purchases") – the cost of new purchases (in US dollars); x3 ("Margin") – the average value of trade margins (in percentages); x4 ("the duration") – the duration of the sale of this drug (in years). The universal sets for the described variables are defined as fol-lows: 1 0;600000U ; 2 20000;1500000U ; 3 0;50U ; 4 0;10U . The uni-versal set for the predicted value coincides, obviously, with 1U .

For each input and for output variables, term-sets are constructed: 1 "small" "medium" "large" "critical" S,M, L,C A ; 2 "small" "medium" "large" S,M,L A ; 3 "small" "medium" "large" S,M, A С ; 4 "short""medium", "long lasting" S,M, L A ; "small" "medium" "large" "critical" S,M, L,C D

Based on the information provided by the expert, the following membership functions are constructed for the terms of input and output variables (Figs. 1–4).

10 0

00

100

000

150

000

200

000

250

000

600

000

1M C В К

80 0

00

100

000

150

000

700

000

800

000

1 20

0 00

0

1C В М

500

000

1 50

0 00

0

Fig. 1. Membership function of linguistic

variable "balance" Fig. 2. Membership function of linguistic variable

"new purchases"

10 15 40

1C В М

30 503 20 3 5 10

1 C Д К

8 122 6 Fig. 3. Membership function of linguistic

variable "margin" Fig. 4. Membership function of linguistic variable

"the duration"


The next step is to build a fuzzy knowledge base (Table 2).

Table 2. The basis of the fuzzy knowledge of the problem

Incoming combination

number (logical rule)

Input variables Weight Output variable

x1 x2 x3 x4 w y

11 М М М К 1

d1

12 В М М С 0,9 13 В М М Д 1 14 К М М С 1 15 М М В Д 0,7 16 М С В С 0,7 17 С М С С 0,8 18 С М С Д 0,8 19 С С С С 0,7 20 В М С Д 0,9 21 В М В Д 0,5 23 С М В Д 1

d2

24 М С С С 1 25 М В В С 0,8 26 С С М С 0,8 27 С В М В 0,7 28 С В С С 0,7 29 В С С С 0,7

d3

30 В С С Д 0,5 31 В С В Д 0,3 32 К С В Д 0,8 33 М В С С 0,9 34 М В С К 1 35 С В М К 1 36 С В С Д 0,8 37 М В В С 0,8

d4 38 М В В Д 0,9 39 С В В С 0,9 40 С В В Д 1 41 В В В Д 1

The following calculations were made for the trading company data for the 2011 sales season 1 80000

x ; 2 36000 x ; 3 22

x ; 4 9 x . In this case, the variable x1

refers to the terms "medium" (with the degree of affiliation 1( ) 1 6000030000

x x )

or "high" (with the degree of affiliation 1( ) 7000030000

x x ); x2 – the term

"small" (with the degree of belonging 1( ) 1100000

x x ); – to the terms "average"


(with degree of affiliation) or "high" (with degree of affiliation); 3x – to the term "long-

term" (with the degree of belonging 1( ) 64

x x ). By simple comparison it is easy

to see that the active rules will be rules 18, 20, 21 (lead to the output 1d ) and 23 (leads to the output 2d ).

The quantitative value of the output value (the result of defuzzification) was calcu-lated by the method of the center of gravity (16). Finally, the value of the forecast value

of the balance of the commodity mass 353865180 26041, 513588, 5

y is equal to that

which is sufficiently close to the real balance of the goods 23200 by the end of the 2016 season, namely US dollars.

Linking the obtained value y y* to the efficiency E, and the predicted value of the output variable with the function of the goal Z, on the basis of formula (4), one can conclude that the value of the firm compared with the previous year is zero in absolute units, since the target function has not been achieved . And according to formula (5), the turnover of the company is about 88%, compared with the previous year.

It should be noted that the predicted values can be more closely approximated to the actual observed (taking into account data for several previous years) by reviewing the established weight coefficients, adjusting membership functions, etc. You can also increase the number of input quantities. However, with their too large numbers, the construction of a fuzzy knowledge base about the unknown dependence becomes a difficult task. This is due to the fact that in the memory of the average statistician at the same time can hold no more than 7 2 the notions-signs. In such cases, it is expedient to carry out the classification of input variables and, in accordance with it, construct a derivation tree that defines the system of embedded statements in each other [11, 15].

One can propose a different approach to processing large volumes of fuzzy data under conditions of incomplete certainty of the vector of input variables (primary char-acteristics). The essence of the approach is based on conducting a simulation of the behavior of the investigated system and an expert assessment of the addition of the existing knowledge base to the new informative data and the establishment of the vector of input characteristics. Obviously, such an approach is iterative and it is necessary to take care of the convergence of the calculation process to achieve the goal with minimal cost and limited error.

When dealing with non-physical data in artificial intelligence problems, the con-struction of recognition systems, expert systems, medical and parametric diagnostics, the creation of logical-linguistic models, the most successfully adapted declarative pro-gramming languages, which in the language of logical statements and functional-logical dependencies provide the opportunity to describe the problem with fuzzy formulated data and obtaining solutions in the form of logical sequences, new functional depend-ences or probabilistic characteristics with definite mathematical By hope and dispersion of the input sign. Ultimately, the initial vector of primary attributes should be refined, which will ensure reliable processing of fuzzy data.

Therefore, the processing of incomplete or unclear information is, on the one hand, the application of the theory of blurry logic (in particular, the theory of fuzzy sets and measures) and the construction of logic trees or, on the other hand, the formation of logical rules from the functional-logical dependencies with indeterminate variables,


which can take both deterministic and probabilistic values. In the process of processing a priori information and selecting appropriate criteria of likelihood, you can replenish the insufficient data, providing the formation of new knowledge.

Obviously, for implementing the described approach to calculating the value of in-formation, it is advisable to use declarative programming languages (Lisp, Prolog or their modifications depending on a specific objective problem) [15], which are most successfully adapted for the functions of the form (4)–(7), (12), (13). These languages can be both analytical and descriptive (functional, logic rules, fuzzy sets), which allows solving problems associated with qualitative recognition and analysis of objects of complex structure (handwriting recognition, handwriting, psychophysiological state of a person , construction and analysis of storage, processing, information security, automat-ed theorizing, environmental monitoring and decision-making) and implementation of the target functions relevant to a specific applied or scientific task.

3. CONCLUSIONS

The authors propose a new approach to assessing the value of information. This approach is based on the theory of pattern recognition, which extends the scope of its application and can be successfully implemented using declarative programming languages [15] or universal language simulation (UML) [2]. The features of its application and directions of further improvement are noted.

The authors cover three main approaches to assessing the amount of information and assessing its value: classical, algorithmic and figurative. There are given a comparative description of these approaches, considered the advantages and limitations of each of them, and the prospects for their use.

Approaches to the processing of fuzzy information under conditions of incomplete definition of the vector of input characteristics based on the theory of fuzzy sets and fuzzy measures, construction of membership functions and application of declarative programming languages are proposed.

The further development of the proposed approaches can be achieved by conduct-ing specific statistical studies related to the need to assess the quantity and value of information based on a figurative approach.

The proposed approach is advisable to use in applied problems, where the mathe-matical description is difficult or completely impossible. This approach will contribute to the development, in fact, of the methods of recognition theory and identification, and the theory of information and coding.

BIBLIOGRAPHY [1] Brillouin L., 1960. Science and Information Theory. Moscow. [2] Gamma E., Helm R., Johnson R., Vlissides J., 2012. Wzorce projektowe (in Poland). [3] Hartley R.V., 1959. Information Theory and its Applications. Moscow. [4] Kharkevich A.A., 1960. About the value of information. Problems of Cybernetics

4, 53–57, Moscow.


[5] Kolmogorov A., 1965. The Three Approach to the Definition of the Concept of "amount of information". Problems of Information Transfer 1(1), 63–67.

[6] Konysheva L.K., Nazarov D.M., 2011. Fundamentals of the theory of fuzzy sets. Piter.

[7] Novak V., Perfiliev I., Mockor I., 2006. Mathematical Principles of Fuzzy Logic. Fizmalit.

[8] Partiko Z.V., 2001. The figurative concept of information theory. Lviv: LNU ім. I. Franko (in Ukraine).

[9] Rothstein O.P., 1994. Differential diagnosis of coronary heart disease on the basis of fuzzy logic. In: O.P. Rottshtein, M.O. Zhupanova, V.M. Sheverda (eds.), Herald VPI 3, 32–38.

[10] Shannon K., 1963. The Works on the Theory of Information and Cybernetics. Moscow.

[11] Syavalko M., 2011. Mathematics of hidden possibilities. Ostrog: Publishing House of the Nauk Stroh Academy.

[12] Turksen I.B., 2002. Fuzzy expert systems. In: M. Zheleny (ed.), St. Petersburg: Peter.

[13] Zadeh L.A., 1973. Outline of a New Approach to the Analysis jf Complex Systems and Decision Processes. IEEE. Trans. Syst. Van and Cybern. 1, 28–44.

[14] Zaiats V.M., Rybitskay O.M., 2013. The hidden possibilities of mathematics for the statistical processing of information. Materials International scientific and technical conf. "System Analysis and Information Technologies" (SAIT-2013), May 27-31, NTUU "KPI", Kiev, 317–318.

[15] Zayats V.M., Zayats M.M., 2016. Logical and functional programming. Tutorial. The Ministry of Education and Science of Ukraine (the list № 1/11-16135 for 13 October 2014) provide Griff, Kamyanets-Podilsky: "Ruta", 400 p. (in Ukraine).

[16] Zimmermann H.J., 1991. Fuzzy Set Theory and it’s Applications. Kluver Academ-ic Publishers, Dordrecht, Boston, MA, 2nd ed.

ANALIZA STATYSTYCZNA SZACUNKU ZNACZENIA I ILOŚCI INFORMACJI W USŁUDZE MASOWEJ

Streszczenie

Na podstawie teorii obrazów proponuje się nowe podejście, które pozwala na ilo-ściowe określenie znaczenia i ilości informacji. Podejścia do definiowania i oblicza-nia podstawowych pojęć teorii informacji, w szczególności ilości informacji i oceny jej znaczenia, oparte są na rozważaniach statystycznych (podejście klasyczne), teorii algorytmicznej (podejście algorytmiczne) i teorii wzorca uznanie (podejście oparte na figuratywności). Podejścia do przetwarzania informacji rozmytych proponowane są w warunkach niepełnego wyznaczania wektora charakterystyk wejściowych; oparte są na teorii zbiorów rozmytych i miar rozmytych. Przeprowadzono ich anali-zę, określono granice ich stosowania oraz obszary skutecznego stosowania, w szczególności w odniesieniu do systemów usług masowych. Słowa kluczowe: ilość informacji, znaczenie informacji, zbiory rozmyte i logika

rozmyta, systemy masowej usługi



SPATIAL ANALOGUES OF NUMERICAL QUASICONFORMAL MAPPING METHODS FOR SOLVING

IDENTIFICATION PROBLEMS

Andrii Bomba, Mykhailo Boichura

Rivne State Humanitarian University – Department of Computer Sciences and Applied Mathematics

Plastova 31 str., 33028, Rivne, Ukraine

Summary. The algorithm of applied quasipotential method for image reconstruc-tion, according to which the alternate application of numerical method for qua-siconformal mapping and solving of parametric identification problems is general-ized to space. The results of numerical experiments are presented.

Keywords: applied quasipotential tomography, quasiconformal mapping, regular-ization, identification, numerical methods

1. ANALYSIS OF RECENT RESEARCHES AND PUBLICATIONS

About the current state of the problem of restoring the picture of conductivity co-efficient (CC) distribution inside the research object using electrical impedance tomog-raphy (EIT) is referred, for example, at [6, 11, 13, 16]. This visualization method (in compare with others) is characterized by non-invasiveness, low cost and non-radiation exploitation, although it does not allow to receive high-resolution images. This, in par-ticular, related to the mathematical aspects of restoring the conductivity value using EIT, which are discussed, for example, in [6, 10, 11, 15]. In general, the problem of restoring the distribution of CC inside the bounded body using a series of measurements at the border both foreign and domestic scientists are engaged (see, for example [1, 4, 6, 8, 10, 11, 13, 15, 16]). In particular, the applied quasipotential method for image recon-struction, according to which, solving of the sequence of so-called analysis and synthe-sis problems is reduced to the alternate application of numerical method for quasicon-formal mapping and parameter identification of conductivity function of the media, respectively. On this basis, a methodology for step-by-step correction of the CC value using, in particular, a specially developed procedure for solving the corresponding sys-tem of linear algebraic equations is developed. On the other hand, the mathematical statement and the algorithm for solving the problem of quasiideal flow modelling in a curvilinear parallelepiped, bounded by the equipotential surfaces and the flow surfaces is given in [3, 5].

26 Andrii Bomba, Mykhailo Boichura

2. PROBLEM STATEMENT

Consider the quasiideal movement processes of particles (for example, electrical charges) in the simply connected curvilinear body (rod – tomographical object) G (Fig. 1a), bounded by limited smooth closed surface ( , , ) :G x y z

( , ) ( , ), ( , ), ( , ) ,r u v x u v y u v z u v 0 1,u 0 1 ,v generated by the potential difference ( )*

p and *( )p *( ) ( )*( 0),p p which are given on a chosen equipotential

surfaces * ** *p p p pA B B A and * *

* * ,p p p pC D D C where ( , ),x u v ( , ),y u v ( , )z u v – defined con-tinuously differentiated functions; * ,pA * ,pB * ,pC * ,pD * ,pA

* ,pB * ,pC

*pD – marked

points on the ;G 1, 2,...p – number of injection (see, for example, [8, 9]); * *

* * ,p p p pA D D A * *

* * ,p p p pB C C B * * * *p p p pA B C D and * * * *p p p pA B C D – impermeable boundary

flow surfaces. We model the current injections through the tomographical object with using, in particular, different pairs { ( ),PKu ( )},PKv corresponding to which

( ), ( ) , ( ), ( )PK PK PK PKPK x u v y u v ( )P K for all possible values from the set of curves * *{ ,p pPK A B * * ,p pB C * * ,p pC D * * ,p pA D * * ,p pA B

* * ,p pB C * * ,p pC D

* * ,p pA D *

* ,p pA A *

* ,p pB B *

* ,p pC C *

* }.p pD D We denote the corresponding surface bound

G with given eight marked points by ( ) .pG

As a mathematical model of applied quasipotential tomography (shunt model) [10], like as [1, 3–5], we use the systems of differential equations in partial derivatives, which connect mutually quasicomplex conjugate quasipotentials ( ) ( ) ( )p p of den-sity ( ) ( )pgrad and flow functions ( ) ( ) ( ),p p ( ) ( ) ( )p p having CC

( ) identification condition and corresponding boundary conditions:

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( ),( ) ( ) 0, ;

p p p

p p p

grad grad gradgrad grad G

(1)

* *( ) ( )* ,p p p p

p pA B B A

* *( ) *( ) ,p p p p

p pC D D C

* *( ) 0,p p p p

pA D D A

* *( ) ( ) ,p p p p

p pB C C B

Q

( ) 0,p p p p

pA B C D * * * *

( ) ( ) ;p p p p

p pA B C D

Q

(2)

*

( ) ( )( ) ( ) ( ) ;

p p p p

p pp p p

A B A A

Q Q ds ds Qn n

* *( ) ( ) ,p p p p

p pA D D A

M M

* *( ) ( ) ,p p p p

p pB C C B

M M

* *( ) ( ) ,p p p p

p pA D D A

M M

* *( ) ( ) ,p p p p

p pB C C B

M M

* *( ) ( )** ,p p p p

p pA B B A

M M

* *( ) **( ) ,p p p p

p pC D D C

M M

Spatial analogues of numerical quasiconformal... 27

* *( ) ( )** ,p p p p

p pA B B A

M M

* *( ) **( ) ,p p p p

p pC D D C

M M

( ) ( ) ,p p p p

p pA B C DM M * * * *

( ) ( ) ,p p p p

p pA B C D

M M

( ) ( ) ,p p p p

p pA B C DM M * * * *

( ) ( ) .p p p p

p pA B C D

M M (3)

Here:

( ) ( ) ,p

p p

A M

M M ds

( ) ( ) ,p

p p

B M

M M ds

( ) ( )** * ,

p

p p

A M

M M ds

**( ) *( ) ,p

p p

D M

M M ds

( ) ( )** * ,

p

p p

A M

M M ds

**( ) *( ) ,p

p p

D M

M M ds

( ) ( ) ,p

p p

A M

M M ds

*( ) ( ) ;

p

p p

A M

M M ds

n – unit vector of outer normal; M – point of the respectively surface section;

( ) ( ),p M ( ) ( ),p M ( ) ( ),p M ( ) ( ),p M ( )* ( ),p M *( ) ( ),p M ( )* ( ),

p M *( ) ( ),p M ( ) ( ),p M

( ) ( ),p M ( ) ( ),p M ( ) ( )p M – functions, constructed by

interpolating experimentally obtained their values ( ) ,pi ( ) ,pi

( ) ,pi ( ) ,pi ( )* ,pi ( ) ,pi

( )* ,

pi

*( ) ,pi ( ) ,pi

( ) ,pi ( ) ,pi

( )pi at ( ) ( )( , ),p pi iu v

( ) ( )( , ),p pi iu v ( ) ( )( , ),p pi iu v

( ) ( )( , ),p pi iu v ( ) ( )* *( , ),

p pi iu v

*( ) *( )( , ),p pi iu v ( ) ( )*( , ),

p pi iu v

*( ) *( )( , ),p pi iu v ( ) ( )( , ),p pi iu v

( ) ( )( , ),p pi iu v ( ) ( )( , ),p pi iu v

( ) ( )( , )p pi iu v points on the respective sections

( ) ( )*( ,

p pi

( ) ,pi ( ) ,pi ( ) *( ) ;p pi

( ) ,pi ( ) ,pi ( )* ,pi *( ) ,pi ( )* ,pi *( ) ,pi ( ) ,pi

( ) 0;pi

( )**0 ,

pi

**( ) ,pi ( ) ,pi

( ) ( ) ;p pi Q ( )0 ,pi ( ) ,pi ( )** ,pi

**( ) ( ) );p pi Q

( ) ( , , ):pG ( ) *( )* ,p p ( )0 ,pQ ( )0 ;pQ

( ) ,pQ ( )pQ

– corresponding redistribution of discharges ( )pQ of the vector fields

(current values) along the horizontal and vertical unit layers of arbitrary cross-sections [3, 5].

We find conductivity coefficient as:

, ,(0 )

( ,...) ,s

a x y z

(4) where:

, ,a (0 )s – the parameters that are being searched during the problem solving.


3. THE IDEA AND AIM OF THE STUDY

We have aim to extend the applied quasipotential method for image reconstruc-tion, proposed in [1] to the spatial case [2]. Like as [1], it is assumed, that CC is given in the polynomial form. The algorithm for solving such problems, as a rule, involves iterative alternating fixation of some parameters and on their basis, search for others. Based on these considerations, the sequential identification of the CC is carried out using the so-called synthesis problem. Its solution is found by minimizing the functional of the sum of the squares of the discrepancy between the desired conductivity function and the posteriori data about the execution of the Cauchy-Riemann type conditions. This, in particular, allows step by step to identify the CC using a system of linear alge-braic equations with arbitrary precision mechanisms type [9]. Solving the analysis prob-lem, we proposed by using method of numerical quasiconformal mapping [1–5]. Last one causes the replacement of above problems for quasiconformal mapping of physical domains into corresponding domains of complex quasipotential by inverse problems.

4. PRESENTATION OF THE MAIN MATERIAL

Algorithms for numerical solving of inverse nonlinear boundary value problems on quasiconformal mappings in curvilinear quadrilateral domains bounded by flow lines and equipotential lines are proposed in [2], and in [3, 5] such approaches are general-ized to space. Accordingly, solving the problem will be carried out applying the given methods (using the corresponding notations; in this regard, algorithms obtained in the above-mentioned works will be fragments of wider structures, in particular, when it is necessary to take into account injectivity), (Fig. 1).

a) b)

Fig. 1. Tomographical object G (a) and corresponding complex quasipotential surfaces

( )pG (b)

We reconstruct the CC using condition for minimizing of functional:


2(1) ( ) ( )1

( ,...) ( ,...) ( ) min,p

p p

pf

(5)

where: (1) (1) ( )

0,0,0 1,0,0 0,1,0 0,0,1 , , ,0,0 0, ,0 0,0,( ,...) ( ,..., , , , , ,..., ,..., , , );p

s s sa a a a a a a a

( ) ( ) ( )( )( ) / .

p p ppf

Using necessary condition for an extremum of function with many variables we have following relations for the formation of the corresponding system of linear alge-braic equations:

2 2 2 2 2 2( ) ( ) ( ) ( ) ( ) ( )( ), ,1 (0 ) 1

( )p p

p p p p p pp

p s pa x y z f x y z

(6)

relative to variables , ,a 2 2 2(0 , ).s The corresponding difference analogues in mesh domains ( )pG when

( ) ( ) ( ), , , ,( )p p p

i j k i j k are written similarly to [1–5].

Here: ( ) ( ) ( ) ( ) ( ), , , , , , , ,( , , ) ;p p p p pi j k i j k i j k i j kG x y z

( ) ( ) ( ) ( ) ( ), , ( , , ) :p p p p pi j k i j kG ( ) ( ) ( ) ,p p pi i ( )=0, +1;pi m ( ) ( ) ,p pj j

( )=0, +1;pj n

( ) ( ) ,p pk k ( )0, 1;pk l

( ) ( )

( )( )

1,

p p

pp

m

( )

( )( )

1,

p

pQp

n

( )

( )( )

1;

p

pQp

l

( )

( ) ( )( ) = ;

p

p pp

( ) ( ) ( ), , ;p p pm n l N

( )p – quasiconformal invariants [2, 3, 5] for corresponding bodies;

We approximate functional (5) and function (4) (and especially the ( )( )pf ap-pearing in (5)) by the same principle as in [1].

Note, that the formulation and solving the system of linear algebraic equations, when it is built on the basis (6) and when 2,s similarly to [1], requires the use of mechanisms for arbitrary precision (for example, type [9]).

Algorithm for solving the problem is based on the rotational parameterization of internal nodes of the grid regions ( ) ,pG CC and using the ideas of block iterative method [1–5, 14]. But unlike [1, 4] we set the number of injections ,p bounds of the domains ( )pG and input, output sections of particles movement through

( )pG (using functions ( , ),x x u v ( , ),y y u v ( , ),z z u v ( , ),вх вхx x u v ( , ),вх вхy y u v

( , ),вх вхz z u v ( , ),вих вихx x u v ( , ),вих вихy y u v ( , )),вих вихz z u v quasipotentials ( )* ,p *( ) ,p partition parameters ( ) ,pm ( ) ,pn ( )pl of the domains ( ) .pG


Then we calculate ( ) *( ) ( ) ( )*( ) / ( 1),p p p pm ( ) ( ) ( )/ ( 1),p p pQ n

( ) ( ) ( )/ ( 1)p p pQ l

and the values of quasiconformal invariants ( ) ( ) ( ) ( )/ ( ).p p p p

Specifying of the local velocity values ( )pi of particles and potentials ( )p

i at the points ( ) ( )( , )p pi iu v (results of physical measurements), we build corresponding depend-encies of their distributions at input sections and at flow surfaces, in other terms

( )* ( )

p and ( ) ( , )p u v by way of interpolation. Then we find the coordinates of angular

points * ,pA * ,pB * ,pA

*pB on ( )pG and discharges ( ) ,pQ

( )pQ

using equations

* ** *

( ) ( ) ( )* *( ) ( ) ,

p p p p

p p p

A B A B

Q ds ds * *

* *

( ) ( ) ( )* *( ) ( ) ,

p p p p

p p p

A A B B

Q ds ds

( ) ( )( ) ( ) *( ) ( )

*( )

( 1)( 1) ( )1

p pp p p p

p

n lQ Qm

[3, 5]. After it we calculate

( ) *( ) ( ) ( )*( ) / ( 1),

p p p pm ( ) ( ) ( )/ ( 1),p p pQ n ( ) ( ) ( )/ ( 1),p p pQ l

quasiconformal invariants ( ) ( ) ( ) ( )/ ( )p p p p and node coordinates ( )0, ,0 ,

pj

( )( )0, , 1

,ppj l ( )0,0, ,

pk ( )

( )0, 1,p

pn k

( 1, ,p p ( )0 1,pi m ( )0 1,pj n ( )0 1)pk l using a difference representations of formulas, inversed to (2), (3). Then we build itera-tive approximations of characteristic points of the flow lines using gradient methods and having a given equipotential surfaces * ** *p p p pA B B A (using direct and reverse moves).

Using this we find the corresponding coordinates of nodes ( ), ,0 ,p

i j ( )( ), , 1 ,pp

i j l

( ),0, ,

pi k

( )( ), 1,pp

i n k

( 1, ,p p ( )1 1,pi m ( )0 1,pj n ( )0 1),pk l and hence points

* ,pC * ,pD * ,pC

*pD on ( ) .pG Nodular lines inside the input and output sections are

constructed on the basis of their particular "quasiorthogonality", as well as using formu-

las *

( )( ) ,

p j

pp

jA M

dsn

*

( )( ) ,

p j

pp

jA M

dsn

*

( )( ) ,

p k

pp

kA P

dsn

*

( )( )

p k

pp

kB P

dsn

(optional here is the requirement for the satisfy of the orthogonal-

ity conditions, that given in [3, 5], instead of it are considered certain procedures of "quasiorthogonality" to ensure the existence of "horizontal" and "vertical" unitary layers of arbitrary cross sections).

The initial approximations of ( ,0), ,p

i j k and parameters (0)

, , ,a that defining CC, in the simple cases of given convex domains, we form, for example, as follow:

( ) ( ) ( )( ,0) ( ) ( ) ( ) ( ) ( ) ( ), , 0, , ,0, , ,01, , , 1, , , 1

( ) / 6p p pp p p p p p pi j k j k i k i jm j k i n k i j l ( )(1 ,pi m ( )1 ,pj n

( )1 ,pk l 1, ),p p (0)0,0,0 1,a (0)

, , 0a (1 ).s After that we start the iterative reconstruction process consists of following steps: for a given injection number 1 p p finding the coordinates of internal nodes using a difference representation of


Laplace type equations [3, 5] (with consider injectivity) when ( )1, ,pi m ( )1, ,pj n ( )1, ;pk l solving the system of linear algebraic equations that it is built on the basis (6)

relative to ( ), ,la (here 0,1,...r – iterative step number, (0 )s ); if

,p p then change the injection number p to 1p and return to refine the coordi-nates of internal nodes; otherwise – we check the conditions for finishing the iterative process, among which may be [2]: the stabilization of the near-boundary nodes, the CC, the quasiconformal degree parameter, the discharge values, etc (1 ,p p ( )1 ,pi m

( )1 ,pj n ( )1 ).pk l In the case where one of these conditions is not satisfied we continue the iterative process, otherwise – we build the corresponding reconstructed image and, – if necessary, – the electrodynamical meshes, the complex quasipotential domains or calculate the current density fields, etc.

It should be noted, that here, as in [1] (as opposed to [2, 3, 5]) is no necessary to use formulae to refine the boundary nodes (since their coordinates are known a priori), which, of course, accelerates the reconstruction process of image. In order to use the machine time more frugally, it is also possible to apply formulas (5) and (6) only for selected points.

To improve the accuracy of quasiconformal mappings (first of all, around the an-gular boundary lines), it is possible to introduce a specially developed procedure for “replace” smoothness to orthogonality of the corresponding flow and equipotential surfaces.

Also note, that instead the procedure for determining the coordinates of the bound-ary nodes using formulas, inversed to (2), (3) (by interpolating the results of physical measurements), we can immediately select them so that the local differences in the values of the function of flow or potential between them at the corresponding neighbor-ing points to be constant within the injection.

We represent the results of numerical calculations for the following input data: we consider that the investigated surface consists of two parts

3 2( , ) 40 60 10,x u v v v 3 3 3 2 2 3 3( , ) 390 660 720 270y u v u v u v u v u v 2 2 3 2 2 2990 360 270 540 180 30 30 20,u v uv u v uv uv v v 3 3( , ) 270z u v u v 3 2 2 3 3 2 2 2 2 2300 270 30 360 90 180 180 30 30 10u v u v u v u v u v uv uv v v and

3 2( , ) 40 60 10,x u v v v 3 3 3 2 2 3 3( , ) 285 960 360 675y u v u v u v u v u v 2 2 3 2 2 3 21440 45 1080 270 30 225 180 150 20,u v uv u v uv v uv v v ( , )z u v 3 3 3 2 2 3 3 2 2 3 2 2135 510 270 375 810 135 540 180u v u v u v u v u v uv u v uv

245 90 90 10uv v v (0 , 1),u v 12,p ( ) ,pm ( ) ,pn ( ) ,pl 3,s (0)0,0,0 1,a (0)

, , 0a (1 ),s ( )* 0,p *( ) 1,p * * ( ),p pA Bu * * ( ),p pA Bv * * ( ),p pC Du

* *( ),

p pC Dv * * ( ),

p pA Bu * * ( ),

p pA Bv * * ( ),

p pC Du * * ( ),

p pC Dv *

*( ) ,

p pA Au *

*( ),

p pA Av

**

( ) ,p pB B

u **

( ),p pB B

v **

( ),p pC C

u **

( ),p pC C

v **

( ),p pD D

u **

( ),p pD D

v ( ) ,pi ( ) ,pi ( ) ,pi

( ) ,pi ( )* ,

pi

( ) ,pi ( )* ,

pi

*( ) ,pi ( ) ,pi

( ) ,pi ( ) ,pi

( ) .pi We obtained results of CC

distribution (Fig. 2b) in comparison with the given theoretically (Fig. 2a).


a) b)

Fig. 2. CC distribution: exact solution (when 0 1,4589336614,a 1 0,0024400653, a

2 0,0063507608, a 3 0,0000054861,a 4 0,0000041829,a 5 0,0000377299,a

6 0,0000000225,a 7 0,0000000064, a 8 0,0000000936,a 9 1,4589336614,a

10 0,0024400653, a 11 0,0063507608, a 12 0,0000054861,a 13 0,0000054861,a

14 0,0000041829,a 15 0,0000377299,a 16 0,0000000225,a 17 0,0000000064, a

18 0,0000000936,a 19 0,0000001641) a (а); approximated solution (б)

Note, that the developed algorithm is sufficiently open for various additions, gen-eralizations, etc., in particular, if the process is unstable, then it is possible, as in [1, 4, 6, 10, 13, 15, 16], to "install" a regularization procedure of Tikhonov type when the func-tional (5) is represented in the form:

211

,... ,...p

p p

pf

(1) ( ) 0,0,0 1,0,0 0,1,0 0,0,1 , , ,0,0 0, ,0 0,0,[ ,..., , , , , ,..., ,..., , ,, , ] min,p

s s sa a a a a a a a (7)

where: (1) ( )

0,0,0 1,0,0 0,1,0 0,0,1 , , ,0,0 0, ,0 0,0,[ ,..., , , , , ,..., ,..., , , ]p

s s sa a a a a a a a – stabilizing

functional; – regularization parameter.

5. CONCLUSIONS AND PERSPECTIVES FOR FURTHER

STUDIES

We proposed the algorithm for reconstruction the inhomogeneous conductivity co-efficient using the applied quasipotential tomographic data based on algorithm of identi-


fication CC contained in [1, 4], and mathematical statement and the method of solving the problem of modeling of the quasiideal stream in a curvilinear parallelepiped bound-ed by the equipotential surfaces and the stream surfaces known from [3, 5]. According to this approach, we are building the spatial electrodynamic meshes and applying the methodology for sequential identification of the CC alternately. This is realized by gradually applying the numerical quasiconformal mapping methods and minimizing the functional of the sum of the squares of the discrepancy between the desired conductivity function (given in the polynomial form) and the posteriori data about the execution of the Cauchy-Riemann type conditions. Last allows the synthesis problem to be reduced to solving the system of linear algebraic equations. We note that if manage to represent the CC as a polynomial, then we have a significant gain in speed of the algorithm (which, in particular, provides the possibility for comparing the relevance and accuracy of the solutions in future researches), however, the presence of at least one term in an

appropriate sum, for example, 1

xx

( 0)x type, significantly increases the time of

solving the problem. Furthermore, essential feature of developed algorithm is the approximately high

speed of its execution, the possibility it’s paralleling and stopping the computing proce-dure when some conditions for finishing the process are complete. Also those areas of the physical domain, where the errors of calculation exist are automatically defined. This allows to use the machine time more frugally. The last is essential, particularly, in the cases of having, so-named, “stagnant zones” and the “zones of large gradients”, which appears near the especial angular lines of unsmooth boundary surfaces of respec-tively bodies.

It is also worth noting that, unlike the traditional approaches to the statement and solving the problem of electrical impedance tomography [6, 8, 10, 11, 13, 15, 16], we set the distribution of the fluid (current) and the averaged potential in the contact areas between plate and body. This certainly provides greater mathematical rigor, and hence the accuracy of the results.

We plan to extend the proposed algorithm to the cases of convective-thermo-diffusion transfer of impurities in the body [7, 12], cases of having a several applied quasipotential sections of output flow. The latter, in particular, will allow for a more precise statement of the problem of applied quasipotentials tomography in comparison to [1–5, 6, 8, 10, 11, 13, 15, 16] due to smaller number of averaging and simplifications.

BIBLIOGRAPHY [1] Bomba A.Ya., Boichura M.V., 2017. Applied quasipotenrial method for solving

coefficient problems of parametric identification. NUWEE, Rivne. [2] Bomba A.Ya., Kashtan S.S., Pryhornytskyi D.O., Yaroshchak S.V., 2013. Com-

plex analysis methods: monograph. NUWEE, Rivne. [3] Bomba A.Ya., Klymiuk Yu.Ye., 2014. Mathematical Modeling of Spatial Singu-

lar-Perturbed Processes of Filtration-Convection-Diffusion Type. Assol, Rivne. [4] Bomba A.Ya., Kroka L.L., 2014. Numerical Methods of Quasiconformal Map-

pings for Solving Problems of Identifying of Electrical Conductivity Coefficient in an Applied Potential Tomography. RSHU, Rivne.


[5] Bomba A., Terebus A., Gavryluk V., 2011. Nonlinear Filtration Processes Mathe-matical Modeling in Free Boundary Spatial Mediums. TNTU, Ternopil.

[6] Brazovskii K.S., 2016. Methods and Technical Means for Evaluating the Func-tional State of the Human Brain based on Electrical Measurements. Doctor of en-gineering thesis, Tomsk.

[7] Chernukha O.Yu., Dmytruk V.A., 2013. Mathematical Modeling the Steady-State Processes of Convective Diffusion in Regular Structuresunder Mixed Boundary Conditions. IAPMM, Lviv.

[8] Gavrilov S.V., 2013. An iterative method for solving a 3D electrical impedance tomography problem in the case of piecewise constant conductivity and several measurements on the boundary. SRCC of MSU, Moscow.

[9] Hanrot G., Lefévre V., Pélissier P., Théveny Ph., Zimmermann P., 2017. GNU MPFR. The Multiple Precision Floating-Point Reliable Library. Available at: www.mpfr.org/mpfr-current/mpfr.pdf.

[10] Holder D., 2005. Electrical Impedance Tomography. Methods, History and Appli-cations. Institute of Physics, London.

[11] Horb M.S. and Husieva O.V., 2013. Choice of the Study Object for Mathematical Model in Electrical Impedance Tomography. NTUU "KPI", Kyiv.

[12] Ivanchuk N., Martynyuk P., Tsvetkova T., Michuta O., 2017. Mathematical mod-eling and computer simulation of the filtration processes in earth dams. Technolo-gy centre, Kharkiv.

[13] Liu D., Kolehmainen V., Siltanen S., Laukkanen A.-M., Seppänen A., 2015. Esti-mation of Conductivity Changes in a Region of Interest with Electrical Impedance Tomography. American Institute of Mathematical Sciences, USA.

[14] Ortega J.M., Rheinboldt W.C., 2000. Iterative Solution of Nonlinear Equations in Several Variables. Society for Industrial and Applied Mathematics, Philadelphia.

[15] Sushko I.O., Rybin O.I., 2014. Visualization of Surface Conductivity Distribution by Conductivity Zones Method. NTUU "KPI", Kyiv.

[16] Tyler T.N., 2015. Conductivity-Based Nanocomposite Structural Health Monitor-ing via Electrical Impedance Tomography. PhD Thesis, Michigan.

ANALOGI PRZESTRZENNE METOD NUMERYCZNYCH

ODWZOROWAŃ QUASIKONFOREMNYCH DO ROZWIĄZYWANIA PROBLEMÓW IDENTYFIKACJI

Streszczenie

Algorytm metody przyłożonych quasipotencjałów do rekonstrukcji obrazów, zgodnie z którym przeprowadza się kolejne zastosowanie metod numerycznych odwzorowań quasikonforemnych do rozwiązywania problemów identyfikacji pa-rametrycznej, uogólniono na przestrzeń. Przedstawiono wyniki eksperymentów numerycznych. Słowa kluczowe: tomografia przyłożonych quasipotencjałów, odwzorowania qua-

sikonforemne, regularyzacja, identyfikacja, metody numeryczne



INTELLECTUAL WIRELESS DIAGNOSIS OF LINEARLY

DISTRIBUTED OBJECTS IN REAL TIME

Gennady Krivoulya, Vladislav Shcherbak

Kharkiv National University of Radio Electronics – 14 Nauky Avenue, 61166, Kharkiv, Ukraine

Summary. Intellectual diagnostics of linearly distributed objects using wireless sensor networks is proposed. The solution of the task of functional diagnostics is realized by the expert system on the basis of the knowledge base in the form of a neuron-fuzzy network. For a technical object the current values of the diagnostic parameters are measured by wireless sensors. As an example, an expert diagnostic system for assessing the operability of a technical object. Keywords: linear wireless network, expert system, diagnostic parameters, sensors,

neural network

1. ANALYSIS OF SUBJECT AREAS

One of the most important tasks to ensure the required level of reliability and safety of complex technical objects is monitoring their current state. In this case, it is necessary to implement the collection, accumulation, processing and analysis of data from spatially distributed sensors in order to detect deviations of controlled parameters from the required value. Observation of a change in the situation allows a timely deci-sion on the possibility of further exploitation of the facility. For complex distributed objects, it is advisable to use wireless sensor networks (WSN), which help you not only measure the values of the monitored parameter (passive sensors), but also manage the processes in objects using active sensors (activators). The lack of wires makes it possi-ble to use the WSN in hard-to-reach places or on mobile objects, which greatly expands the range of sensor networks [1].

Many of the application areas of the WSN are related to the positioning of sensors in the form of a linear structure, which leads to the emergence of a new type of WSN, which can be defined as linear sensor networks LWSN. Such structures include rail-ways, pipelines for oil, gas, water, etc. objects that can have a length of hundreds or even thousands of kilometers. The use of LWSN for monitoring tasks has some fea-tures that are related to a linearly distributed structure and a significant length of the network. In particular, such networks require considerable time for serial data transmis-sion, have high power consumption while reducing the reliability of the transmission of information [2].

36 Gennady Krivoulya, Vladislav Shcherbak

To eliminate the above disadvantages, LWSN and unmanned aerial vehicles (UAVs) are used to monitor linear infrastructures that collect and transmit data. At the same time, the end-to-end transmission of data in the network is reduced, its reliability and fault tolerance are increased, the service life of the sensor batteries increases, the required quality of service is provided.

Structurally, the monitoring system has four types of nodes: sensors for collecting information (SN), relay nodes (RNs), UAVs and data receivers. SN nodes use the clas-sic routing method to transfer their data to the nearest RN, which acts as a cluster gate-way for surrounding SNs. The UAV moves sequentially along a reciprocating trajectory along a linear network and transmits the data that is collected by the RN to receivers located at both ends of the LWSN [7].

The use of UAV implements the advantages of the linearity of such networks to increase reliability, efficiency, energy savings and network lifetime [7].

Figure 1 shows the architecture of the monitoring system for LWSN. This system architecture has three levels. Wireless ground and underground sensors represent the bottom layer of the architecture, which provides the highest degree of detail for moni-toring. At the second level, multimedia means and sensors increase the accuracy of the system by collecting visual information. At the third level, mobile terrestrial robots and UAVs represent the highest level of information gathering and provide the necessary completeness of diagnosis [6].

The communication between the RN nodes and the receivers is provided by the UAV, which moves between the receivers and collects the necessary data from the RN. The UAV board computer can also perform other functions such as data aggregation, scheduling, route, touch operating system and software configuration, programming, updating, and synchronizing SN and RN nodes. The UAV can be used to transport data and programs from receivers to SN, and may also have GPS capabilities. Since the through delay for transmitted data depends on the network structure and the UAV route, the on-board computer can analyze various options for reducing this parameter [6].

The main amount of information in the monitoring system is present in the first lower layer and with a large number of sensors there is a problem large data processing (Big Data). At the same time, the promising direction in the process of creating intelli-gent systems of functional diagnosis is the use of hybrid neural networks, which com-bine the advantages of fuzzy expert systems and neural networks. False logic in the development of the knowledge base (KB) and the mechanisms of output allows to for-malize a procedure for assessing the technical condition on the basis of unreliable and inaccurate information when identifying possible malfunctions [3]. For the formation of logical conclusions in the intelligent diagnostic system (IDS), knowledge is used in the form of fuzzy function with linguistic variables that are represented by terms with some membership function (MF). In the presence of automated technical means for storing and collecting information from sensors of the object of diagnosis (OD) there is an opportunity to automate the replenishment of the KB and track the huge volumes of changing information, to take quality and timely solutions when diagnosing complex technical objects [4].

Intellectual wireless diagnosis of linearly... 37

Fig. 1. Monitoring system for linearly distributed objects

Today, smart grid, smart homes, smart water networks, intelligent transportation, are infrastruc-ture systems that connect our world more than we ever thought possible. The common vision of such systems is usually associated with one single con-cept, the internet of things (IoT), where through the use of sensors, the entire physical infrastruc-ture is closely coupled with information and communica-tion technologies; where intel-ligent monitoring and management can be achieved via the usage of net-worked em-bedded devices. In such a sophisticat-ed dynamic system, devices are interconnected to transmit useful measurement information and con-trol instructions via distributed sensor networks.

A wireless sensor network (WSN) is a network formed by a large number of sen-sor nodes where each node is equipped with a sensor to detect physical phenomena such as light, heat, pressure, etc. WSNs are regarded as a revolutionary information gathering method to build the information and communication system which will greatly improve the reliability and effi ciency of infrastructure systems. Compared with the wired solu-tion, WSNs feature easier deployment and better fl exibility of devices. With the rapid technological development of sensors, WSNs will become the key technology for IoT. The use and evolution of WSNs within the wider context of IoT, and provide a review of WSN applications, while also focusing the attention on infrastructure technologies, applications and standards featured in WSN designs.

Monitoring of complex distributed engineering facilities requires the application of a measuring and information system in the form of a complex of devices for collecting information on the status of a monitored facility. Engineering objects have a complex structure with dynamic internal processes, so to reflect the current state at specified times it is necessary to have a parallel measurement of its parameters. At the present stage of the development of cybernetic systems, the solution of this problem can be effectively implemented on the basis of WSN). Currently, sensor networks are undergo-ing a stage of active implementation and search for new applications, one of which is the monitoring of various parameters in real time. The use of the WSN allows the crea-tion of a dispatch system that provides the operator with continuous access to infor-mation on the status of the serviced objects. The need to remove information about the

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